Problem Statement
Construct two tangents to a circle from a point outside the circle.
Given Data:
- Circle with centre O and radius 3.0 centimeters.
- Point P located 7.0 centimeters from the centre O.
To Construct:
Two tangents PA and PB from the external point P to the circle.
Step-by-Step Construction
Follow the animation on the left.
Why does this work?
This construction relies on the property of the angle in a semicircle.
1. We treat the line segment OP as the diameter of a new imaginary circle.
2. The angle formed in a semicircle is always ninety degrees (a right angle).
3. Therefore, the angle between the radius OA and the line PA is ninety degrees.
4. A line perpendicular to the radius at the point of contact is a Tangent.
Examiner's Advice
Do not erase construction arcs!
Marks are awarded for showing the method (the arcs), not just the final lines.
- Sharp Pencil: Use a hard pencil (2H) for construction lines and a softer pencil (HB) for the final tangents.
- Labelling: Always label points (O, P, A, B) immediately after marking them.
- Arrows: Put arrowheads on the tangent lines to indicate they are lines, not just segments.
- Dotted Lines: Use dotted lines for the perpendicular bisector to distinguish it from the main figure.